Make the cake: Preheat the oven to 350 degrees F. Lightly grease a 10-inch Bundt pan and dust with flour. In a large bowl, sift the 1 3/4 cups of cake flour with the baking powder, salt, and 3/4 cup of the sugar. Stir in the shredded coconut. In another large bowl, whisk the egg yolks with the coconut milk, oil, and vanilla.

In the bowl of a stand mixer fitted with the whisk, beat the egg whites until frothy, 1 to 2 minutes. With the mixer on, gradually add the remaining 1/2 cup of sugar and beat until the whites are glossy and soft peaks form, 1 to 2 minutes.

Make a well in the center of the dry ingredients. Mix in the coconut milk mixture until incorporated. Stir one-fourth of the beaten egg whites into the batter, then carefully fold in the remaining beaten whites until no streaks remain. Scrape the batter into the prepared pan.

Bake the cake for 45 to 50 minutes, until golden and the edges start to pull away from the pan. Transfer to a rack and let cool for 1 hour. Run a paring knife around the side of the cake to loosen it, then invert the cake onto a plate.

Make the frosting: In a medium bowl, whisk the confectioners' sugar with the coconut milk until smooth. Spoon the frosting over the cooled cake, allowing it to drip down the side. Garnish with toasted shredded coconut.